2^(k-p) Fractional Factorial Design
Also known as: 2^k-p design, fractional factorial, screening design, Kesirli Faktöriyel Desen (2^k-p Fractional Factorial)
The fractional factorial design is an economical experimental strategy that investigates k factors by running only a carefully chosen 1/2^p fraction of the full 2^k factorial experiment. Formalized by George E. P. Box and J. Stuart Hunter in their landmark 1961 Technometrics paper, it exploits the sparsity-of-effects principle — that high-order interactions are typically negligible — to screen many factors with far fewer runs than a complete factorial would require.
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When to use it
Use a fractional factorial design when you have many factors (typically five or more) to screen and resources permit only a fraction of the full factorial. The critical assumption is the sparsity-of-effects principle: higher-order interactions (three-factor and above) are negligible. For Resolution III designs, main effects must be interpretable in isolation from two-factor interactions. The response should be continuous and approximately normally distributed, and the error variance should be homogeneous across runs. A minimum of eight runs is required; more runs increase the resolution and the ability to estimate independent effects.
Strengths & limitations
- Dramatically reduces the number of experimental runs compared with a full factorial, making large screening studies feasible.
- The aliasing structure is fully known through the defining relation, so the experimenter can choose a design that confounds only negligible effects.
- Directly extendable: important factors identified in a screening fraction can be followed up with augmented designs (e.g., fold-over, response surface) that resolve aliases.
- Well-supported by classical statistical theory with known power and confidence-interval formulas.
- Aliasing means that a significant effect cannot be unambiguously attributed to a single source without additional runs or prior knowledge.
- The sparsity-of-effects assumption may fail in systems where interactions are intrinsically important.
- With very few runs (e.g., Resolution III in 8 runs), there is no degrees of freedom for pure error unless center points or replicate runs are added.
- Factor levels must be set at exactly two levels per factor in the basic 2^(k-p) form; quantitative optimisation of factor settings requires a subsequent response-surface study.
Frequently asked
How do I choose the resolution for my design?
Resolution III is sufficient when you only need to identify the most active main effects and are willing to assume all two-factor interactions are negligible. Resolution IV keeps main effects clear of two-factor aliases — a safer choice for most screening studies. Resolution V additionally lets you estimate two-factor interactions independently, at the cost of more runs. Select the lowest resolution that is scientifically justifiable for your application.
What is the defining relation and why does it matter?
The defining relation is the complete set of aliased effect chains implied by your choice of generators. For example, choosing the generator I = ABCDE means the main effect A is aliased with BCDE. Before running the experiment you should write out all alias chains and confirm that no important effects are confounded with each other; if they are, switch to a different generator set or a higher-resolution design.
Can I follow up a fractional factorial with more runs?
Yes — this is the standard strategy. A fold-over augments a Resolution III design by adding the mirror-image fraction, dealiasing all main effects from two-factor interactions and yielding a Resolution IV design. A foldover of a Resolution IV design separates the aliased two-factor pairs. Alternatively, adding axial and center points converts the fraction into a central composite design suitable for response-surface modelling.
What if I have fewer than eight experimental units?
Below eight runs a 2^(k-p) design cannot maintain even Resolution III for more than a few factors. In this situation consider Taguchi orthogonal arrays, which can accommodate three-level factors and noise factors, or reduce the number of factors to those with the strongest prior evidence of importance.
Sources
- Box, G.E.P. & Hunter, J.S. (1961). The 2^(k-p) Fractional Factorial Designs. Technometrics, 3(3), 311–351. link ↗
- Montgomery, D.C. (2017). Design and Analysis of Experiments (9th ed.). Wiley. ISBN: 978-1119492443
How to cite this page
ScholarGate. (2026, June 1). 2^(k-p) Fractional Factorial Design. ScholarGate. https://scholargate.app/en/experimental-design/fractional-factorial
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